Find the composition.
If f(x) = x^2 + 12 and g(x) = x^2 - 5, find (g*f)(x)
f(x) = x^2 + 12 and g(x) = x^2 - 5, find (g*f)(x)
g(f) = (x^2 + 12)^2 - 5
= x^2 +24 x +144-5
= x^2 + 24 x +139
(g*f)(x) or (g○f)(x)
=g(f(x))
= g(x^2+12)
= (x^2+12)^2 - 5
= x^4 + 24x^2 + 139
To find the composition (g * f)(x), we need to perform the function g on the function f. In other words, we will substitute f(x) into g(x).
Given that f(x) = x^2 + 12 and g(x) = x^2 - 5, we can substitute x^2 + 12 into g(x):
(g*f)(x) = g(f(x))
= g(x^2 + 12)
Now, substitute x^2 + 12 into g(x):
= (x^2 + 12)^2 - 5
To simplify this further, we can expand the equation:
= (x^4 + 24x^2 + 144) - 5
= x^4 + 24x^2 + 144 - 5
= x^4 + 24x^2 + 139
Therefore, the composition (g * f)(x) is given by (x^4 + 24x^2 + 139).